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Theory of phase segregation in DNA assemblies containing two different base-pair sequence
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2017 New J. Phys. 19 015014
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New J. Phys. 19 (2017) 015014
https://doi.org/10.1088/1367-2630/aa5482
PAPER
OPEN ACCESS
Theory of phase segregation in DNA assemblies containing two
different base-pair sequence types
RECEIVED
26 September 2016
REVISED
28 November 2016
ACCEPTED FOR PUBLICATION
Dominic J (O’) Lee1, Aaron Wynveen2 and Alexei A Kornyshev1
1
2
Department of Chemistry, Imperial College London, SW7 2AZ, London, UK
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
19 December 2016
E-mail: [email protected], [email protected] and [email protected]
PUBLISHED
Keywords: DNA, homology recognition, phase segregation, statistical mechanics, DNA assemblies
30 January 2017
Supplementary material for this article is available online
Original content from this
work may be used under
the terms of the Creative
Commons Attribution 3.0
licence.
Any further distribution of
this work must maintain
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and DOI.
Abstract
Spontaneous pairing of homologous DNA sequences—a challenging subject in molecular biophysics,
often referred to as ‘homology recognition’—has been observed in vitro for several DNA systems. One
of these experiments involved liquid crystalline quasi-columnar phases formed by a mixture of two
kinds of double stranded DNA oligomer. Both oligomer types were of the same length and identical
stoichiometric base-pair composition, but the base-pairs followed a different order. Phase segregation
of the two DNA types was observed in the experiments, with the formation of boundaries between
domains rich in molecules of one type (order) of base pair sequence. We formulate here a modified
‘X–Y model’ for phase segregation in such assemblies, obtain approximate solutions of the model,
compare analytical results to Monte Carlo simulations, and rationalise past experimental observations. This study, furthermore, reveals the factors that affect the degree of segregation. Such
information could be used in planning new versions of similar segregation experiments, needed for
deepening our understanding of forces that might be involved, e.g., in gene–gene recognition.
1. Introduction
In this article we develop a statistical mechanical model of DNA assemblies containing mixtures of fragments of
two different base-pair sequences that are allowed to phase segregate. This model, e.g., may describe phase
segregation of DNA in cholesteric spherulites. Spherulites are liquid crystal structures formed in DNA-noncondensing electrolytic solutions upon applying mild osmotic stress that helps to overcome net repulsion
between the DNA molecules. This current study was initially motivated by experiments [1] in which partial
segregation of DNA molecules, according to their base-pair sequence texts (i.e., the order of their nucleic bases),
was observed in spherulites containing a mixture of two DNA species with different sequence texts.
These experiments were performed by mixing an equal amount of 294 bp-long fragments of synthetic
dsDNA with two different sequences in 0.5 M NaCl under mild osmotic stress. Each of the two types of
sequences contained the same numbers of GC and AT base pairs, but the base pairs were shuffled randomly into
two different orderings. As osmotic pressure was applied slowly, causing the molecules to condense into
spherulites, the two sequence types simultaneously segregated, which was observed using confocal microscopy.
The fragments were observed to be in a cholesteric arrangement with a large pitch (a small tilt angle between
cholesteric layers), and so the arrangement of molecules could be described as being, roughly, in a columnar
phase, in which DNA fragments are arranged parallel to each other. To detect segregation between the two
species in these experiments [1], each species was labelled by chromophores of different colour (red or green). In
order not to overload the samples with chromophores, only 5% of DNA fragments were thus labelled. But this
fraction was sufficient to see a signature of homology segregation, in which DNA fragments with the same base
pair ordering preferentially condensed in domains richer in their proportion. This was indicated by the
appearance of red and green patches in the observed images for each spherulite. The degree of colour segregation
© 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
was significant enough that it could not be explained simply by random statistics. Conversely, control
experiments in which dsDNA fragments of only one species, half labelled with one colour chromophore and half
with the other, demonstrated random mixing of the molecules labelled with the two chromophore types. These
ruled out segregation effects due to the type of chromophore, and, hence, a difference in base-pair sequence was
shown as the primary driving force behind the segregation.
Such evidence of homology recognition in a protein-free environment has been widely discussed in the
scientific press [2] as being an innate consequence of the structure of DNA. Signatures of homology recognition
were also reported earlier in gel electrophoresis studies [3]; and some time later, strong indications of the same
effect were obtained by single molecule manipulation techniques [4] and AFM studies for DNA on nucleosomes
[5]. Further corroborative evidence of homology recognition in a protein free environment was presented in a
very recent study, in which preferential interaction of long homologous tracts, in parallel orientation within
single molecules, was observed [6] in magnetic bead distance-force experiments.
The existence of such interaction forces that distinguish between similar and different molecules might be
biologically important. Indeed the pairing of chromosomes of similar sequences is an essential step in many
biological processes [7–9]. It has been speculated whether this initial pairing might be some innate general
characteristic of DNA [9], resulting, again, from the contrasting forces between DNA with differences in
sequence.
Prior to the work of [1], it was shown that the phenomena of phase segregation between different DNA
sequences might indeed be theoretically possible. Indeed, an explanation of the results of osmotic stress
experiments presented in [1] relies on the idea that the overall interaction energy between each fragment,
although net repulsive, should contain a helix-specific attractive component. The degree of attraction may be
modelled quantitatively through the Kornyshev–Leikin (KL) theory [10, 11]. In the theory, the attractive
component is found to be stronger between homologous molecules than between two non-homologous
fragments upon considering the non-ideal helix structure of DNA molecules [12, 13]. This difference, referred to
as the ‘recognition energy’, was first calculated in [14]. The theory of homology recognition was then developed
with a higher level of sophistication in a series of subsequent works (for review see [10, 15], and for a recent study
[16] with citations to later articles contained therein). The recognition energy can comprise several kB T per
persistence length, and it increases with the length of the molecules. However, in the segregation experiments of
a liquid-crystal configuration [1], the employed molecules had to be sufficiently short—not much longer than
the dsDNA bending thermal persistence length—to form well-ordered liquid crystalline phases. Those
molecules used in [1] were roughly twice the persistence length, and according to theory, segregation effects
would have been weak. This, indeed, was observed, but distinguishably seen.
Later works have suggested that the preferred conformation of DNA with the same sequence will be parallel
(i.e. parallel alignment with base pair sequences running in the same direction) to each other but not antiparallel
(parallel alignment but base pair sequences in opposite directions), as this recognition depends on the
homologous sequences being in face-to-face register, with base pairs of the same type for the two molecules
being directly across from each other [16].
Alternately, preferential pairing between identical DNA sequences might be, in some part, due to
interactions that depend locally on the type of base pair content [6, 17, 18]. Here, the interaction energy between
two like base pairs has lower interaction energy than two non-alike ones. One such mechanism, mediated by
counter-ions being localised by particular base pairs [19], has been discussed in [6]. An alternate mechanism
utilising secondary hydrogen bonding was proposed in [17]. Sequence specific variations in van der Waals
interactions might also be responsible [18]. Such mechanisms would also be influenced by the base-pair specific
pattern of DNA distortions and are manifestly helix-structure-dependent forces. Though, in this work, we
assume that the global pairing mechanism due to helix distortions (discussed in the previous paragraph) is the
root cause of segregation.
In this paper, we lay the groundwork in developing a model that may rationalise the degree of segregation
under different conditions. In the main text we develop a statistical mechanical description taking into account
key details of the putative helix specific forces in DNA–DNA interactions; in the supplementary material we
append a simpler phenomenological model which we will refer to in the main text for comparison. In fact, the
final form of the partition function that we use in our calculations (see the end of next section) is independent of
a specific pairing mechanism, up to the choice of model parameters, as it simply relies on the fact that there are
helix-structure-dependent forces.
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
Figure 1. Schematic of the crossections of two DNA molecules in parallel alginment at the midpoint of their long axes. The diagram
shows azimuthal orientations, f1 and f2, of the two molecules, the relative positions of both the major and minor grooves, as well as
the phophate strands. Also shown are counter-ions adsorbed at the molecular surface. The arrows that bisect the minor grooves define
these azimuthal angles, which we also refer to as the molecules’ ‘spins’.
2. Model
2.1. General considerations
Below, we formulate the principles of the theory that could describe phase segregation between two types of
DNA fragments, each with differently ordered base pair texts, in liquid crystalline columnar assemblies.
Our theory takes into account thermal fluctuation modes corresponding to rotations of the molecules about
their long axes. These modes are important if the helix structure of DNA molecules in DNA–DNA interactions
matters. Indeed, for interactions that depend on helix structure, the pair interaction between molecules depends
on the relative azimuthal orientation of two molecules about their long axes [10]. Thus, such fluctuations are
restricted by helix dependent interactions.
Any pair potential (for two molecules labelled 1 and 2) that depends on the continuous helical symmetry of
the molecule, for molecules aligned parallel to each other, may be expanded as a Fourier series in f1 - f2
¥
V1,2 (R) = L å a n (R) cos (n (f1 - f2)) ,
(2.1)
n= 0
where f1 and f2 are the (‘spin’) angles that both helices (at the midpoint of the long axes of the molecules) make
to an axis that lies in a plane perpendicular to the long axes of both the molecules (see figure 1). We use the term
‘spin’ to emphasise an isomorphism (or analogy) between this approach for modelling interactions in columnar
DNA assemblies and a kind of 2D X–Y model (modified to account of an extra cosine(s)) of magnetism. In the
latter, the angle describes the orientation of real, magnetic spins. In equation (2.1), R is the inter-axial separation
between the two molecules. Also, in writing equation (2.1), the interaction has been assumed to scale linearly
with the length of the molecules, L. This supposes the two molecules are fixed to be in complete parallel
juxtaposition with each other. In a 2D-assembly of N molecules, there are N such spin angles fm describing
these degrees of freedom and N (N - 1) 2 pair potentials.
Our model describes a single layer of DNA molecules, in columnar alignment, whose molecular centres are
assumed to be fixed to a 2D regular lattice structure, in a plane perpendicular to the molecular centre lines, as in
[20]. Though, as in [20], we allow for the lattice to distort from a hexagonal to a rhombic structure to minimise
the free energy in response to the DNA–DNA interactions. As in [20], as well as summing up the pair potentials
for the lattice, we truncate the sum in equation (2.1) at n = 2. Thus, our theory involves a modified version of an
X–Y magnetic model, originally studied in [20].
Reference [20] solely dealt with columnar aggregates comprising only of one type of DNA fragments, i.e., all
of them having the same sequence of base pairs. However, here, with the intention of describing the experiments
of [1], we extend this model to the case where there are now DNA molecules with two different sequence texts.
For there to be phase segregation between molecules with different sequences, there must be a more attractive
interaction between like (homologous) molecules than between different (non-homologous) ones, as was
suggested by previous works [14–16, 21]. Here, the difference in the interaction between homologous and nonhomologous molecules results from the helical distortions, specifically in the twist and rise between adjacent
base-pairs, of the DNA double helix. The pattern of distortions is dependent on base pair text [22]. Molecules
with the same base pair sequence share the same distortions, and therefore can maintain an energetically
preferential charge alignment along their full length, whereas non-homologous molecules, with different
sequences and, therefore, different distortions, cannot maintain this alignment. These patterns of distortions
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
influence the positions of the negatively charged helical phosphate groups along the molecule as well as the
positions of the groove centres, where positively charged counter-ions may localise. Therefore, homologous
pairs may possess a lower interaction energy than their non-homologous counterparts. This was shown to be the
case with the mean-field electrostatic KL-theory [14, 15].
To model helical distortions of the molecules, modifying the model considered previously [20], we replace
the constant azimuthal (or, again, ‘spin’) angle fm with fm (z ) (for a molecule at site m in the assembly), since the
‘spin’ angle depends on the position z along the long axes of the molecules due to the distortions of the helical
twist of the molecules. (A discussion and justification for this can be found in [10, 14, 15].) Hence, the function
fm (z ) depends on the base pair sequence of the molecule.
The form for the interaction energy is then obtained by replacing L in equation (2.1) with an integral over z
along the length of the molecule. As there are DNA molecules of two different sequences, and thus two different
distortion patterns, the azimuthal orientation for a molecule at the lattice site m are labelled either as fmA (z ) or
fmB (z ); with the molecule types A and B. An additional degree of freedom in the system is thus manifested since
we allow for the assembly to exchange molecules of type A and B with an external reservoir, and thus allow for
the relative proportions of the two types of molecule in the assembly to vary.
Summing up all the pair potentials, we can write the total interaction energy for the assembly (including all
pair interactions) as
Eint =
1
2
L/ 2
⎡ a (∣R - R ∣) - a (∣R - R ∣) cos (f (z ) - f (z ))
1
n
m
n
⎣ 0 m
m
n
ò-L/2 dz må
¹n
a
+ a2 (∣Rm - R n∣) cos (2fam (z ) - 2f nb (z ))⎤⎦ .
b
(2.2)
As mentioned, in this expression, we have truncated the sum over helical harmonics (equation (2.1)) at n = 2 (as
in [20]). The position vectors Rm and Rn are restricted to fixed lattice sites m and n. Here, the indices a and b can
either be set to A or B, depending on which of the two species of molecules occupies the sites m and n,
respectively.
Now, the azimuthal angles fam (z ) (and fnb (z )) can be expressed as
fam (z ) = f0m +
1
h
ò0
z
d Wa (z ¢) dz ¢ ,
(2.3)
where, h (3.4 Å) is the average spacing between two adjacent base-pairs along a molecule. Here f0m is the
azimuthal angle that the DNA double helix associated with the lattice site m at its molecular centre makes with a
single axis (defined throughout the assembly); the function dWa (z ) represents the pattern of distortions away
from an ideal helix, which, again, is dependent on the base pair structure. For an ideal helix dWa (z ) = 0. If two
neighbouring molecules possess random sequences with respect to each other, i.e., are non-homologous, the
two functions dWa (z ) are uncorrelated over large length scales along the DNA molecules, and therefore obey
Gaussian statistics:
ád W A (z ) ñW = ád WB (z ) ñW = 0,
h2
d (z - z ¢) ,
l(c0)
ád W A (z ) d WB (z ¢) ñW = ád WB (z ) d W A (z ¢) ñW = 0.
ád W A (z ) d W A (z ¢) ñW = ád WB (z ) d WB (z ¢) ñW =
(2.4)
In equation (2.4), the subscript W of the averaging brackets refers to ensemble averaging over all possible base
pair realisations.
The parameter l(c0) = (1 lW, W + 1 lh, h - 2 lW, h )-1 is the (intrinsic) helical coherence length [22] of
DNA molecules, where lW, W and lh, h are the contributions due to variations from the mean values of,
respectively, twist angles and base pair rises (vertical distances) between adjacent base-pairs along any DNA
molecule. The correlation length lW, h arises from a correction due to correlations between the rise and the twist
angle. One can represent lW, W = h ád W2ñ and lh, h = h (g 2 ádh2ñ ) (here g = 2p H , where H is the average
helical pitch of DNA), where ád W2ñ and ádh2ñ are effective standard deviations of base pair twist and rise, between
adjacent base pairs, away from their average values.
Thus, l(c0) is a measure of the level of distortion away from the ideal, undistorted, helix structure. For an ideal
double helix, l(c0) = ¥ . In general, then, the difference in the interaction energies between homologous and
non-homologous molecular pairs only becomes pronounced when the molecules are longer than l(c0). This is
effectively the length over which the two non-homologous sequences can stay commensurate—strands on one
molecule aligned with grooves on the opposite molecule—with each other (if not allowing for torsional
flexibility), and maintain helix structure dependent interactions with each other. Hereafter, we use the value
l(c0) » 150 Å, which is within the range estimated in [22].
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
Note that fam (z ) also depends on thermal torsional fluctuations of the molecules. For short molecules this
simply renormalizes l(c0) [10]. However, this can be neglected, as it is a relatively small correction. Another, less
trivial effect is torsional adaptation where the helical structure of the molecules adjusts to facilitate better helix
dependent interactions, due to the elasticity of the molecule [23, 24]. This effect on the pair interaction was
studied in [16, 25–27]). Including these effects in molecular assemblies is mathematically and computationally
involved for finite length molecules [28]. However, here we may, in the first approximation, neglect this effect by
considering molecules that are not much longer than the thermal persistence length of DNA, like those used in
the liquid-crystalline experiments [1].
Using equation (2.3), it is possible to recast equation (2.2) in a more practical form:
Eint =
1
2
⎛
L/ 2
[a 0 (∣Rm - R n∣) - a1 (∣Rm - R n∣) cos fm
ò-L/2 dz må
⎝
¹n
⎜
(0)
- fn(0) +
⎞
(1 - sm s v ) DfW (z )⎟
⎠
2
sm
⎞
⎛
+ a2 (∣Rm - R n∣) cos ⎜ 2f(m0) - 2f(n0) + sm (1 - sm s v ) DfW (z )⎟ ,
⎝
⎠
(2.5)
where
DfW (z ) =
1
h
ò0
z
(W A (z ¢) - WB (z ¢)) dz ¢ .
(2.6)
In equation (2.5), the freedom of choice between type A or B molecules has been characterised in terms of a spinvariable sm = 1; sm = 1 corresponds to type A and sm = -1 to type B.
Thus, the partition function (summing over both sm and f0m degrees of freedom) can be defined as
N
Z=
 å ò0
m = 1 sm =-1,1
2p
dfm0
⎛ Eint {sm , f0 } ⎞
m
⎟.
exp ⎜⎜ ⎟
kB T
⎝
⎠
(2.7)
Potentially, for two specific base pair sequences, one should be able to determine the distortion patterns and
calculate the partition function for those two sequences. However, our task is to calculate a more general,
ensemble-averaged free energy, using the Gaussian statistics described previously. This average is defined as
áF ñW = á- kB T ln Z ñW .
(2.8)
To calculate equation (2.8) precisely would be rather laborious and computationally expensive, involving
detailed simulations that model specific patterns of helical distortions of many possible base-pair realisations,
which then must be averaged. It might be possible to perform some kind of replica trick [29], on equation (2.8),
to facilitate performing the ensemble average. However, this is not without its problems: firstly the ensemble
average still cannot be performed analytically for this type of model, and secondly the analytic continuation in
the number of replicas, n to zero is not a well-defined limit, and particular care must be taken. Instead, we utilise
an approximation in which each pair potential for different helical distortion realizations is replaced with an
ensemble-averaged one. This is effectively replacing quenched disorder with annealed disorder. This
approximation automatically arises within the framework of a variational mean-field approximation, where we
neglect correlations in sm. When the molecules are completely phase segregated or completely mixed those
correlations are unlikely to be important; however, close to the transition such assumption may break down.
The details of the variational approximation are fully presented in the supplemental material, whereas a more
heuristic derivation of it, which relies on assuming annealed disorder, is presented in the main text. Thus, we
write
áF ñW » - kB T ln Z eff ,
(2.9)
where
Z eff =
N
 å ò0
2p
m = 1 sm =-1,1
⎛ áEint {sm , f(0) } ñW ⎞
m
⎟.
df(m0) exp ⎜⎜ ⎟
k
T
B
⎠
⎝
(2.10)
The approximate free energy described by both equations (2.9) and (2.10) is also computed in MC simulations
(for details see section 3 below). One may compute the leading order correction (and higher order corrections)
to equation (2.9) in both simulations and the variational approximation, from considering the full average
equation (2.8). Again, the method used to find these corrections is discussed in the supplemental material, which
relies on an alternate to the replica trick. The average áE int {sm, f(m0) } ñW is simple to obtain:
5
New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
Figure 2. Diagram illustrating the various possible states that the molecules can be azimuthally ordered with respect to their ‘spin’
angles f(m0). In each figure the arrows represent average spin directions (see figure 1). The three possible states here include: (a) the F
state, where all the average spins face the same way; (b) the P state where, around a triangular plaquette, the spins all point in different
directions with the average differences between two spin pairs being the same (more information on this latter state can be found in
[20]); and (c) the AF state where the molecules form rows, in which there are alternating rows of spins in the same direction.
áEintñW =
⎡
⎛
(1 - sm sn ) ⎛ L ⎞ ⎞
L
⎢a 0 (∣Rm - R n∣) - a1 (∣Rm - R n∣) ⎜⎜1 Y ⎜ (0) ⎟ ⎟⎟ cos (f(m0) - f(n0))
å
2 m ¹ n ⎢⎣
2
⎝ 2lc ⎠ ⎠
⎝
⎛
(1 - sm sn ) ⎛ 2L ⎞ ⎞ ⎤
+ a2 (∣Rm - R n∣) ⎜⎜1 Y ⎜ (0) ⎟ ⎟⎟ ⎥ cos (2f(m0) - 2f(n0) ) ,
2
⎝ lc ⎠ ⎠ ⎥⎦
⎝
(2.11)
where the function Y is defined as
Y (x ) = 1 -
1
(1 - exp ( - x )) .
x
(2.12)
The structure of equation (2.11), without the specific form for Y, is quite generic; thus its form may be also
applicable to other candidate mechanisms for homologous pairing . To consider solely a local pairing
mechanism (as described in introduction and also discussed in [6]) one could simply set Y = 1 in
equation (2.11). Independently of this, as often is the case for statistical mechanical studies of specific systems,
the overall model described by equations (2.9)–(2.11) may be of more general interest, as it describes a modified
X–Y spin model describing particles with two types of magnetic interactions.
2.2. Possible phases and transitions in the model
The model potentially allows for many different states. First of all, we may have states of different azimuthal
ordering of the molecules about their long axes. If the ratio a2 (R ) a1 (R ) is sufficiently small between nearest
neighbours, we expect a state witháf(m0) - fn(0) ñ = 0, where, on average, all the azimuthal angles (‘spins’) f(m0)
possess the same value. We call this state the ‘ferromagnetic’ (F) State. But, when the ratio a2 (R ) a1 (R ) becomes
sufficiently large, states for which áf(m0) - fn(0) ñ = 0 no longer holds everywhere may be energetically
preferential. Two such examples are an ‘antiferromagnetic’ (AF) rhombic state and an antiferromagnetic ‘Pott’s’
(P) State. In the simplest of these, the AF state, the molecules in the 2D lattice form rows or 1D layers. Along one
particular lattice direction (within the row), nearest neighbour molecule’s spins are parallel,áf(m0) - f(n0) ñ = 0;
however, the relative angle between neighbouring molecule’s spins along other directions is non-zero,
áf(m0) - fn(0) ñ = y, where y takes a finite value (see figure 2). The P state has a slightly more complicated
structure in terms of f(m0), which is illustrated in figure 2. If we allow for rhombic lattice distortions, the AF state
generally has a lower energy state compared to the P-state, for the values of parameters explored here. However,
the free energies of these states are quite similar [20]. (If we fix the lattice as being triangular (hexagonal), the
P-state will dominate at large values of a2 (R ), since the AF state is unstable without lattice distortions [20].)
Generally, we are interested in determining how phase segregation (separation of non-homologous
molecules) depends on different molecular parameters. Segregation is driven by the size of a1 (R ) Y (L 2l(c0) ) and
6
New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
a2 (R ) Y (2L l(c0) ) for nearest neighbours in the lattice. When both a1 (R ) and a2 (R ) increase, so that the
interaction energy between molecules dominates over mixing entropy, we might expect a phase almost pure in
one type of molecule (complete segregation) as opposed to a mixed state containing significant numbers of both
types. However, these two interaction terms may also compete against each other, and, therefore, lead to nonmonotonic effects in the boundary between the mixed and segregated phases.
The degree of the phase segregation is also controlled by the function Y, which characterises the difference
between the interaction between homologous and non-homologous molecules. Formally, we may expect that
the largest degree of segregation occurs for very long molecules (as L goes to infinity). Here Y takes on its largest
value, meaning the difference in the interaction between homologous pairs and non-homologous pairs is
maximised. Note, however, that when the molecules are sufficiently large, the twisting elasticity (not included in
the current model) would reduce this effect, allowing non-homologous molecules (again, molecules with
different sequence text) to twist about their long axes in such a way to reduce their distortional mismatch, and
hence their interaction energy [26]. On the other hand, for these long molecules, whose lengths are on the order
of several bending persistence lengths, the liquid crystalline structures seen in the experiments of [1], generated
to observe phase segregation, cannot be formed. And hence, we neglect this effect in our study as we desire to
describe segregation effects only of (approximately) columnar liquid-crystalline mesophases of DNA.
2.3. Parameters for the KL model of DNA–DNA interactions
In our numerical calculations, the interaction coefficients, a 0 (R ), a1 (R ), and a2 (R ), are determined from the KL
model for helix structure specific DNA–DNA interactions [10, 11], which is based on a Debye–Bjerrum model
(utilising mean-field electrostatics where ion-ion correlations are neglected). It entails the electrostatic
interactions arising from the helical charge patterns on dsDNA. Here, the double helix is described by two
phosphate backbones separated by two grooves—the smaller, minor groove and the larger, major groove (see
figure 1). Each phosphate bears one negative charge, thus tracing out a double-helical charge distribution.
Adsorbed counterions, cations, are assumed to be localised within these grooves, and therefore are considered as
fixed charges on the molecular surface. These form helical positive charge charge distributions running along the
grooves. All other ions in the system that are not bound to DNA provide linear Debye screening of these helical
charge distributions. Within the framework of this model, the interaction terms a 0 (R ), a1 (R ), and a2 (R ) are
given by
a 0 (R) =
W (k R , k n a )
K 0 (kD R)
2lB (1 - q )2
2l
+ 2B å [xn (q , f1 , f2)]2 n n
,
2
2
(akD K1 (kD a))
(akn Kn¢ (kn a))2
le
le n
a1 (R) =
a2 (R) =
4lB [x1 (q , f1 , f2)]2
le2
4lB [x 2 (q , f1 , f2)]2
le2
(2.13)
K 0 (k1 R)
,
(ak1 K1¢ (k1 a))2
(2.14)
K 0 (k2 R)
,
(ak2 K2¢ (k2 a))2
(2.15)
where
Wn (kn R , kn a) = - åKn - j (kn R) Kn - j (kn R)
j
I ¢j (kn a)
,
(2.16)
kD2 + n2g 2 .
(2.17)
K ¢j (kn a)
and
xn (q , f1 , f2) = qf1 + qf2 ( - 1)n - cos (nf˜ s) ,
kn =
Here, In (x ) and Kn (x ) are modified Bessel functions of the first and second kind, and I ¢n (x ) and K ¢n (x ) are their
derivatives with respect to their arguments, respectively. The parameters of the model are: lB, the Bjerrum length
(lB » 7 Å); le , the mean distance between phosphate charges per unit charge (le » 1.7 Å); kD, the inverse Debye
1
screening length (in all calculations we use the value kD = 7Å ); g = 2p H , where H is the average helical pitch
of DNA (H = 33.8 Å); a is the effective electrostatic radius of DNA (a » 11.2 Å); f̃s is the angular half-width of
the minor groove (f˜s » 0.4p ); q is the degree of counterion condensation (or binding) near the DNA surface (if
the charge of phosphates is 100% compensated with adsorbed counterions, q = 1 ); f1 is the proportion of
condensed (bound) ions localised in the minor groove; and f2 is the proportion of condensed (bound) ions
localised in the major groove.
We will also investigate the more generic phase behaviour of the phase diagram of our system, for a
triangular lattice, in terms of a1 and a2, without their actual R-dependence being specified. Concentrating solely
on these interaction terms realises results that are independent of the underlying interaction model, based on
symmetry arguments presented above.
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New J. Phys. 19 (2017) 015014
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3. Methods
We calculate the free energy of these systems in order to determine when phase segregation occurs via two
means: (i) a variational mean-field approximation and (ii) Monte Carlo simulations. We compare the results of
the two methods to test their accuracy.
3.1. Variational approximation
In the variational approximation, when mixing between the two species occurs, we may use a ‘jellium’-like,
mean-field approximation. Here, we suppose that we can write the following mean-field ansatz (for the
‘ferromagnetic’ state) for the probability that a lattice site m is occupied by either type A or B
Pma  (1 - c) da, A + cda, B .
(3.1)
Here, c = áNBñ N is effectively an order parameter that characterises the degree of phase segregation, where
NB is number of lattice sites which are occupied by molecules of type B . In the model the total number of lattice
sites N is fixed, and they are occupied either by A- or B-molecules, so that always NA + NB = N is constant (NA
is number of lattice sites which are occupied by molecules of type A). Hence, we have (1 - c ) = áNAñ N . Thus,
it is evident that when c = 0 the system contains purely molecules of type A; whereas c = 1 corresponds to
only type B; and when c = 1 2, there is 1:1 mixing of the two species.
Note that the above ansatz is shown to be equivalent to performing a more rigorous variational
approximation using an effective ‘magnetisation’ (see supplemental material). However, using equation (3.1) is
more physically transparent than this more sophisticated method, so we present it in the main text.
In the variational approximation, we may write an approximate free energy
FT = - kB T ln ZT ,0 + áááEintñW ña - ET ñT ,
(3.2)
where
ZT ,0 =
N
¥
 ò-¥
m= 1
⎛ ET {f(0) } ⎞
m ⎟
df(m0) exp ⎜⎜ ⎟,
k
T
B
⎠
⎝
(3.3)
and
ááá Eint ñW ña - ET ñT =
1 N

ZT ,0 m = 1
¥
ò-¥
⎛ ET {f(0) } ⎞
m ⎟
df(m0) ( ááEint {f(m0) } ñW ña - ET {f(m0) }) exp ⎜⎜ ⎟.
k
T
B
⎠
⎝
(3.4)
The variational approximation is chosen to have a Gaussian (or harmonic) trial function where the finite range
of values of f0m is neglected, in both equations (3.3) and (3.4), and the limits of integration are extended to
infinity. Here, the subscript a on one of the averaging brackets represents averaging over the effective probability
given by equation (3.1). In the variational trial function, ET {f0m} are variational parameters that are chosen to
minimise FT to give the most accurate approximation to the full free energy (equation (2.9)).
For an illustration of the approximation method, we will consider in detail the F state. The trial energy
function describing this state is given by
J
ET = kB T å (f(m0) - f(n0) )2 ,
(3.5)
ám, n ñ 2
where the angular brackets on the summation indices indicate that only nearest neighbour interactions have
been considered. Indeed, J, an effective spring constant for relative rotations in f(m0) - fn(0) for nearest
neighbours, is chosen to minimise equation (3.2), which is the closest value to the exact free energy. The
variational approximation can be extended to include next-to-nearest neighbours and so on.
The variational approximation yields, for the F state (considering only nearest neighbours),
⎛ L ⎞⎤
⎛ 1 ⎞⎡
FT
= 3a 0 (R) L - 3a1 (R) L exp ⎜ - ⎟ ⎢1-2c (1 - c) Y ⎜ (0) ⎟ ⎥
⎝ 6J ⎠ ⎢⎣
N
⎝ 2lc ⎠ ⎥⎦
⎛ 2L ⎞ ⎤
⎛ 2 ⎞⎡
k T
+ 3a2 (R) L exp ⎜ - ⎟ ⎢1-2c (1 - c) Y ⎜ (0) ⎟ ⎥ + B ln J + kB TCˆ hex .
⎝ 3J ⎠ ⎢⎣
2
⎝ lc ⎠ ⎥⎦
(3.6)
When we use equation (3.1), in order to optimise c we must also include a mixing entropy to realise a total free
energy
F = FT - TSmix ,
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(3.7)
New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
where
Smix = - kB N [(1 - c) ln (1 - c) + c ln (c)] .
(3.8)
The terms in FT favour segregation of the two molecular types, while mixing entropy Smix opposes this. The
optimal c balances these two effects.
In the more rigorous variational approximation, this additional entropy term is automatically included in
the free energy and does not need to be added by hand (see supplemental material). Further details and how to
formulate the variational approximation for the AF state (using a similar trial function to equation (3.5)) are
given in the supplemental material. For the P state, we have not extended the mean-field approximation
considered in [20] to consider mixing between the two molecule types. This is because the approximation for
molecules of solely one type of base pair sequence (or ideal helices) is already very complicated. This is due to the
reduced lattice symmetry of the P state. Additionally, it does not work very well due to thermal topological and
domain-wall type excitations that were not incorporated [20]. Thus, MC simulations are required to probe the
free energies for a fixed triangular lattice in which this P state occurs.
In appendix A of the supplemental material we present a simpler model, for ease of understanding, as it
captures some key features of the analysis. Most importantly it describes the competition between mixing
entropy and the interaction energy difference between homologous and non-homologous molecules.
Similarities, as well as the disadvantages, of this model compared with the analysis used in the main text are
discussed in the supplemental material.
3.2. MC Simulations
In the computational analysis, before applying the Monte Carlo Algorithm, the initial state of a system, at a given
average nearest neighbour distance, was determined by performing a lattice sum of nearest-neighbour
interactions. Here, the ground state energy was minimised upon varying (i) neighbouring spins of molecules, (ii)
the type of pair interaction, being either of the homologous or non-homologous form , and (iii) the rhombic
angle ω which provides the level of distortion of the rhombic lattice from a triangular one, where w = 60.
The Monte Carlo simulations were carried out on a 24×24 site rhombic or triangular lattice with periodic
boundary conditions for molecules of two different types. Here, only pair interactions (given by the terms in the
summation of equation (2.11)) between nearest neighbours were considered. Three types of Monte Carlo move
were considered: (i) altering the azimuthal angle f(m0) of a molecule at a lattice site, (ii) changing the type
(sequence) of a molecule (for instance, from type A to B) at a lattice site, or (iii) changing the rhombic angle w of
the lattice. Individual simulation runs were performed for systems of molecules with different lengths, densities,
charges, and charge distributions. Furthermore, more generic simulations were carried out varying the
interaction terms a1 and a2.
According to the Metropolis algorithm, MC moves were accepted with a probability of exp (-DE kB T ), if
the energy of the new configuration was DE greater than that of the original configuration, and a probability of 1
if the energy of the new configuration was less than that of the old configuration. Systems were allowed to
equilibrate for 105 MC steps before statistical averages of the states were calculated.
4. Results and discussion
4.1. Phase segregation and consistency between the two methods
In our results, we have determined the position of a mixing transition between a phase-segregated lattice state,
with predominantly one type of molecule in the lattice, and a state in which two types of molecules are evenly
mixed. This has been obtained for a range of interaction parameters for the KL model of DNA–DNA
interactions. The transition line between these two states is defined as where á∣NA - NB∣ñ N = 0.5.
In determining the position of this mixing transition, both methods of calculation, the analytical variational
approximation and the Monte Carlo simulations, agree well (see figure 3), indicating the reliability of these two
approaches. However, each of the methods is not without its minor drawbacks. In the variational calculation, the
system is treated at the mean-field level, thus we cannot expect it to be accurate at the transition point, since
inhomogeneities and correlations in the spatial distribution of the two molecule types on the lattice are likely to
become important here. On the other hand, the MC simulations were carried out on a relatively small lattice of
molecules, and so finite-size effects may influence phase segregation behaviour in these simulations. For
example, one clear artefact of a finite sized lattice, is that á∣NA - NB∣ñ N remains significantly above zero for
values of R ave , the average interaxial distance between molecules, well above the transition. In fact, for the
simulation results above the transition, the value of á∣NA - NB∣ñ N remains constant as R ave is increased. This
constant value, arising from random ordering in a finite system, is calculated to be 2/pN , which, for the lattice
size used in the simulations, is found to have a value of 0.033, consistent with what was found in the MC
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
Figure 3. Phase segregation curves comparing the two methods of calculation. Plotted is á∣NA - NB∣ñ N , the fractional difference of
species (sequence) populations within the lattice. When á∣NA - NB∣ñ N = 1 complete phase segregation between two types of
molecule has occurred, corresponding to a lattice of only one type of sequence. If the value of this parameter equals zero, this
corresponds to perfectly even mixing with the same populations of both species spread over the lattice sites. Results are shown for
molecules with a charge compensation q = 0.7. Here, (a) and (c) are for a molecular length L = 340 Å, whereas (b) and (d) have been
calculated for molecules of length L = 680 Å. In both (a) and (b) a molecular groove charge distribution of f1 = 0.35 and f2 = 0.65
has been used, and in (c) and (d) the molecular charge distribution was given by f1 = 0.5 and f2 = 0.5, i.e., the same amount of
bound charge was apportioned to the minor and majors grooves. Simulation results are provided as points (connected with dashed
lines for clarity’s sake), whereas analytical results are shown as solid lines. Note, for the parameters in (c), the sharp segregation
transition occurs at the location of the F–AF state transition (see figure 4), whereas the segregation transition for the other cases occurs
within the ‘ferromagnetic’ regime (figure 4). Note that for the variational calculationá∣NA - NB∣ñ N = ∣1-2c∣.
simulations. As shown in figure 3, near the transition, the curves for the simulations in figure 3 are much less
steep than those of the variational calculation. But it is well known that finite size effects generally smear
transitions, which is precisely what is observed in the finite-sized MC simulations. We presume that exact results
for á∣NA - NB∣ñ N lie somewhere between these two curves.
This mixing transition, as shown, depends on molecule length and the distribution of charge on the
molecular surface as provided in the interaction model. The effect of increasing length is to push the transition to
larger values of R ave, which thus favours the segregated state over a larger density range in these DNA assemblies.
Indeed, this result is not surprising. According to our model, the difference between the average interaction
energy between homologous and non-homologous DNA increases as the molecular length increases, and hence,
this increasing difference would increase the density range over which phase segregation occurs. Of the two ion
distributions shown in figure 3, we find the distributions with the larger value of f1 ( a greater fraction of ions
localised in the minor groove) yield transitions at smaller interaxial distances R ave. As shown in figure 5, this is a
general trend, which we will discuss below.
4.2. The AF–F transition
Besides the transition between the mixed and segregated phases of molecules with different sequences, there is
also an ‘antiferromagnetic’–‘ferromagnetic’ (AF–F) transition, as discussed in section 2.2. This second transition
changes the ordering of the azimuthal orientations of the centres of the DNA helices about their long axes. At
large values of R ave (whenáf(m0) - fn(0) ñ = 0 for all nearest neighbours), the lowest energy state corresponds to
the F state. As R ave decreases, a2 (in equation (2.11)) that favours áf(m0) - fn(0) ñ ¹ 0 grows faster relative to a1,
which then results in the AF rhombic lattice (or P state for molecules fixed to a triangular lattice) being the lowest
energy state. Again, for the AF state, the molecules are azimuthally ordered in rows. The average value of the
azimuthal angle difference between neighbouring molecules áf(m0) - f(m0) ñ = 0 within a single row of molecules,
while, along other lattice directions, between nearest neighbours (i.e. between rows), áf(m0) - fn(0) ñ = y, where
y is some no-zero value. (Again, for details, see section 2.2.)
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New J. Phys. 19 (2017) 015014
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Figure 4. The averaged ‘spin–spin’ correlation between molecules at neighbouring lattice sites. The figures (a)–(d) correspond to the
same parameter values used to generate those same plots in figure 3. An illustration of the rhombic lattice unit cell (in the top left
panel) is used to identify the two curves (red and green) shown in each of the panels as between different nearest neighbours. The point
of the F–AF transition occurs at the value of R ave, the average lattice separation, where the two curves of the average difference between
the azimuthal orientation (spins) between neighbouring molecules diverge. For large average molecular separations (low densities),
the F state is preferred; whereas, at small separations, as the antiferromagnetic term (a2 (R )) in the interaction grows faster than the
ferromagnetic term (a1 (R )) with diminishing R ave, the AF state is preferred. For increasing values of R ave, in the ‘ferromagnetic’
regime, the spin interaction between molecules grows weaker, and so the spin correlation weakens, i.e., ácos (f(m0) - f(v0) ) ñ  0.
Again, as shown in panel (c), the location of the segregation transition occurs at the same interaxial spacing as the F–AF transition,
whereas for all other cases, the segregation transition occurs within the F state (see figure 3).
To probe this transition, as well as to measure the amount of azimuthal fluctuations in the lattice, we
consider the ‘spin–spin’ correlation function ácos (f(m0) - fn(0) ) ñ between nearest neighbours. As fluctuations
increase, the value of ácos (f(m0) - fn(0) ) ñ decreases. Indeed, for large interaxial spacings between molecules, we
expect both the size of a1 and a2 in equation (2.11) to exponentially decrease (see equations (2.14) and (2.15))
with increasing R ave, leading to a weaker interaction between the molecules and, hence, a decrease in the
‘spin’–‘spin’ correlation function as shown in figure 4. for large R ave .
In the AF state, between the rows of molecules with different ’spins’ (see figure 2), we can write
ácos (f(m0) - f(n0) ) ñ = cos y ácos (f¢m(0) - f¢n(0) ) ñ .
(4.1)
Here, f(m0) - fn(0) = y + f¢m(0) - fn¢ (0) and the values f¢m(0) - f¢n(0) fluctuate about the value zero, i.e.
áf¢m(0) - fn¢ (0) ñ = 0. Indeed ácos (f¢m(0) - f¢n(0) ) ñ is reduced by thermal fluctuations. In equation (4.1), as y
increases, ácos (f(m0) - fn(0) ) ñ must also decrease. Thus, in the AF state, between rows with different ‘spins’ we
expect that ácos (f(m0) - fn(0) ) ñ is smaller than its value along the rows. Due to this difference, in figure 4, the F–AF
transition takes place where the red and green curves of ácos (f(m0) - fn(0) ) ñ, for nearest neighbours in the two
lattice directions, as functions of R ave diverge.
By comparing both figures 3 and 4 we may judge the relative positions of the mixing transition with respect
to the F–AF transition. For the parameter values f1 = 0.35 and f2 = 0.65, the mixing transition occurs well
within the F state of the molecules, whereas, for f1 = 0.5 and f2 = 0.5 (an even distribution of bound
counterions distributed in the minor and major grooves), the mixing transition still occurs well within the F state
for longer molecules (L = 680 Å), but for the smaller molecules (L = 340 Å), the mixing transition occurs at
the same interaxial molecular spacing as the AF–F transition.
Increasing the ratio f1 f2 , i.e., the fraction of bound ions in the minor groove to that in the major groove, has
the effect of pushing the AF–F transition to larger interaxial spacing, Rave since doing so increases the size of the
ratio a2 (R ) a1 (R ) that controls the transition, as it is the relative proportion of the two interaction terms that
dictates the strength of the antiferromagnetic aspect of the interaction. Furthermore, increasing the ratio of
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
Figure 5. Average inter-axial spacing of the segregation transition, RC , as a function of interaction model parameters. In these plots,
the value of L = 1000Å was used. The transition point RC is defined at the value of R ave where á∣NA - NB∣ñ N = 0.5. Shown in (a)
are plots of RC against charge compensation q for two different charge groove distributions and (b) plots of RC against the proportion
of ions in the minor groove, f1 (with f2 = 1 - f1) for two different values of charge compensation.
a2 (R ) to a1 (R ) essentially reduces the strength of the interaction (in the F state) since these terms come into the
interaction with opposite signs. Therefore, the mixing transition moves down to smaller interaxial spacing as the
a2 (R ) a1 (R ) ratio grows, since here, thermal fluctuations grow, leading to increased mixing of different
molecular species. The position of the F–AF transition is insensitive to L. However, increasing the length, L, of
the molecules does significantly reduce the degree of f(m0) fluctuations, as can be seen in figure 4.
The plots shown in figure 4 are found from the MC simulations, where we observe a smooth transition
between the segregated and mixed states. The variational calculation yields a discontinuous AF-F transition (see
[20]), which more or less occurs around where the curves in figure 4 show a more dramatic weakening in the spin
correlation between the molecules. The reasons for the main discrepancies result, again, from the finite size
effects of the simulations or the mean-field elements of the variational theory breaking down very close to the
point of transition.
4.3. The dependence of segregation on KL-model parameters
For further exploration of the effects of changing the counter-ion distribution close to the surface of the
molecules, we specifically analyse results for molecules of a particular (average) length, L = L* , where
L* = 1000 Å or 294 base pairs. This was the length of molecules used in the segregation experiments of [1]. For
these length molecules, we then find the location of the mixing transition location, defined as RC = R ave where
á∣NA - NB∣ñ N = 0.5, upon varying q (the counterion charge compensation of the DNA molecule) and f1 and
f2 (again, the values that denote how bound or adsorbed counterions are apportioned between the minor and
major grooves). These terms were varied because the effective DNA charge and distribution of counter-ions
were found to depend on the species and concentrations of ions used in experiments. In these results we suppose
that f1 + f2 = 1, which should be more or less the case for ions that bind strongly to the grooves.
To describe the experiments of [1], f1 + f2 is not precisely known, however, but it is likely smaller than 1
considering the monovalent salt solutions used in these experiments. However, a DNA charge compensation
value of q = 0.3-0.4 could be predicted for the experiments upon fitting the osmotic pressure data of DNA
osmotic pressure experiments under similar conditions [30, 31]. Finally, these fits revealed that the results were
rather insensitive to the value of f1 + f2 for these smaller charge compensations q. Later we consider a set of,
more or less, realistic values of model parameters to describe those experiments.
In figure 5(a), results are shown for systems in which the DNA charge compensation, q, is varied for two sets
of charge groove distributions, f1 and f2 values, namely f1 = 0.5, f2 = 0.5 and f1 = 0.35, f2 = 0.65. In the case
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
Figure 6. A contour plot showing the degree of mixing between molecules of two different sequences, molecule types A and B, upon
varying the relative strength of the coefficients of the helical interaction between molecules, as well as their overall strength with
respect to thermal energies. The coloured contours give the mixing order parameterá∣NA - NB∣ñ N , where ‘1’ corresponds to fully
segregated and ‘0’ to fully mixed. In addition is included a red ring which corresponds to a plausible range of parameter values for the
experiments of [1] (see main text). Results are shown for simulations with molecules of length L* = 1000 Å .
where f1 = 0.35, f2 = 0.65, the interaxial spacing of the mixing transition, RC , increases strongly with q. The
helical electrostatic coefficient, a1, of the interaction increases as more counter-ions are localised to the helical
grooves (q becoming larger). Therefore, at a given average separation, R ave, this increase results in stronger
preferential interactions between identical molecules, which dominates the thermal fluctuations at separation
values below the mixing transition. It is this that pushes the mixing transition to a larger interaxial spacing RC .
However, when the counterion bound charge is more evenly distributed between the grooves, as seen for the
values f1 = 0.5, f2 = 0.5, the effects of varying q are weakened by the increase in the ratio a2 (R ) a1 (R ). Again,
since a2 (R ) comes in with an opposite sign with respect to a1 (R ) in the interaction, the larger value of a2 (R )
effectively weakens the interaction, and, hence, the impetus for segregation, so that the system remains mixed
until the assemblies are at larger densities (smaller values of R ave ), with the increase of q .
We can also fix q and vary f1 , holding f2 = 1 - f1 , such that counterions are bound either to the minor or
major grooves. In figure 5(b) results are shown for two fixed values of the charge compensation, q = 0.5 and
q = 0.7. In both cases, we find that value of RC depends dramatically on the value of f1 . Again, as more counterion charge is localised in the minor groove ( f1 is increased) the ratio of a2 (R ) a1 (R ) increases, effectively
reducing the difference in interaction energies between homologous and non-homologous pairs. This then
hinders phase segregation, such that, again, RC , the location where segregation occurs, occurs at smaller
intermolecular spacing. In figure 5(b), for the larger value, q = 0.7, we find that the mixing transition occurs at
greater interaxial separations than for the smaller charge compensation, q = 0.5, for most values of f1
considered, due to a reduction in overall interaction strength. However, this difference becomes smaller as f1 is
increased, and reverses at values above f1 » 0.5 . This is again governed by the increasing ratio of a2 (R ) a1 (R ),
as was explained above for figure 5(a).
4.4. The a1 - a2 phase diagram for a fixed triangular lattice
As well as simply focusing on the KL model of interaction, we also show results that are independent of this
model, though relying on the global mechanism of homology recognition through helix distortions. Here we
show the degree of phase segregation as a function of the parameter values a1 and a2 for helix structure specific
interactions. Results are shown, again, for molecules with the length used in the experimental segregation
studies [1].
This figure illustrates some features discussed in the previous sections. Firstly, as the helix-structuredependent interaction increases, (i.e., a1 increases), which occurs with increased aggregate density or charge
localisation within the DNA grooves, the relative difference in the interaction energies between homologous and
non-homologous increases, favouring segregation. Secondly, as more counter-ion charge is apportioned to the
minor groove, increasing the relative value of the antiferromagnetic a2 term to the ferromagnetic a1 term, the
effective helical interaction decreases. Hence, the difference of the interaction energies, again, between
homologous and non-homologous molecular pairs decreases, suppressing phase segregation so that it occurs at
even larger values a1 (and thus, at larger aggregate densities).
Interestingly enough, as the ratio of a2 a1 is further increased, this trend reverses slightly (see the behaviour
of the ‘segregation line’ in right side of the figure 6). Here, the effective helical interaction increases again and,
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
hence, the difference of the interaction energies between homologous and non-homologous molecules. This
effect may be linked to the fact that the ground state here is now the P state. This azimuthal ordering is obscured
by fluctuations, however it still may contribute to free energy, although on average it is not the most popular
state: the mixing transition, for this molecular length, always lies on average in the F state. For this reason this
opposite trend in figure 6 is very weak. Indeed, we would expect for P state that increasing a2 would increase the
effective strength of interactions.
We find that phase transitions between the topologically disordered and ordered spin P-states [20] lie in the
phase segregated region for molecules of chosen length L*. The phase diagram for such spin transitions is
described in [20]. The effect of reducing the length would be to bring this spin transition closer to the phase
segregation line and eventually, crossing it, which might lead to interesting effects, as observed for the shorter
molecules in figures 3 and 4. On the other hand, phase segregation for L = L* was already weak under the
conditions explored in [1].
In the phase segregation transition plot, figure 6, we also show the approximate region in the a1 - a2
parameter space that may correspond to the segregation experiments of [1], using the K-L model of DNA-DNA
interaction. To determine the possible range of values of these parameters, fits were performed on results of
osmotic pressure experiments of DNA aggregates [30, 31] under similar experimental conditions as those found
in [1]. With these fits, estimates of q » 0.3-0.4 could be deduced for the level of charge compensation of DNA
by adsorbed counterions. Appropriate values for f1 and f2 , the fraction of counterions adsorbed in the minor
and major grooves, respectively, could not as easily be deduced from the osmotic pressure experiments.
However, with the relatively small value of the estimated counterion charge compensation found from these fits,
the influence on the interaction terms a1 and a2 of the distribution of counterion adsorption, provided by f1 and
f2 , is rather small. Thus, to limit the number of parameters, we assumed a random counterion distribution along
the DNA molecule, which corresponds to values of f1 and f2 being 0.
A larger influence is exerted on the interaction terms by the separation between neighbouring molecules,
since these terms decrease exponentially with molecular separation. The molecular separations had not been
measured in those experiments. However, since the aggregates in [1] were found to be in a mild cholesteric state,
we could surmise that the average interaxial separation between neighbouring molecules was roughly 3–4 nm,
which corresponds roughly to a surface-to-surface separation of 1–2 nm.
With the estimates for charge compensation q and average molecular interaxial separation (along with the
values of f1 = f2 = 0 ), using equations (2.14) and (2.15), we can determine the possible range of values for the
interaction terms, a1 and a2 that correspond to the conditions of the segregation experiments [1]. Because of,
mainly, the large uncertainty in the distances between neighbouring molecules, there, in turn, was a large
uncertainty in these values, with a1 L* varying anywhere from 0.4-1.4kB T , and a2 varying from 0.05-0.2a1.
This experimental region, drawn in figure 6, straddles the segregation transition boundary of the simulations. It
encompasses the mixing ratio range á∣NA - NB∣ñ N » 0.1-0.5 found, by quantifying the degree of red-green
colour separation, in the experiments. Of course, on the simulation side, there is still a degree of uncertainty in
the precise location as well as the width of the mixing transition since, again, finite size effects of MC simulations
have the tendency to broaden the transition. And, of course, on the experimental side, we have not accounted for
smearing of the transition in experiments due to non-equilibrium effects during the formation of these DNA
mesophases.
5. Concluding remarks
In this article we have used variational approximations and computer simulations to establish the nature of
segregation of DNA molecular fragments in a model for phase segregation of two DNA types. We have shown
how this segregation depends on a variety of molecular parameters that characterise the difference in the
interaction energies between fragments of homologous DNA molecules, sequences with the same base pairs, and
non-homologous molecular pairs, which have different, totally uncorrelated sequences. Here, to follow one
specific line of enquiry, we have assumed that the difference of energies relies exclusively on the base-pair
sequence-dependent patterns of helical distortions away from an ideal helix. This is not to say that there could be
mechanisms relying on local base pair dependent interactions [6, 17, 18] playing a role. The extent of such a role
has yet to be ascertained. However, it should be relatively easy to adapt this general statistical model to take
account of such mechanisms, once analytical models of interaction have been proposed.
As well as phase segregation, our model also predicts different states in which the centres of the molecules
can be azimuthally ordered, which can be described in the statistical mechanical language of magnetic systems
(the helical interaction between DNA pairs depends on the relative azimuthal orientation (or ‘spin’) of the
molecules about their long axes). Indeed, it is quite possible that our model may also have application in the
study of magnetic systems, when two types of magnetic species are considered. Of particular interest to us was
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New J. Phys. 19 (2017) 015014
D J (O’) Lee et al
the relative molecular separation at which the mixing transition between DNA molecules (between the two types
of sequences) occurs with respect to that of magnetic-like transitions of the molecular spin states. Granted,
detection of these ‘spin-ordered’ DNA states may be difficult to detect experimentally, unless synchrotron x-ray
diffraction experiments are invoked. On the other hand, such changes in the azimuthal ordering could affect the
phase segregation of the two molecular types under certain conditions. Generally, though, for realistic model
parameters, we found that this mixing transition occurs when the DNA molecules azimuthal orientations are
aligned on average, i.e., they are in language of magnetic spins in a ‘ferromagnetic state’. However, we discovered
non-trivial effects of the helical interaction on the location of the mixing transition. Namely, the a2 helical
harmonic that favours the antiferromagnetic state in our helical interaction model may reduce the effective
strength of helical interactions. This reduces the difference of the interaction energies between homologous and
non-homologous molecular pairs (the ‘molecular recognition energy’), which, in turn, increases the likelihood
of mixing of molecules of different sequences. Also, we found that if that if the ‘anti-ferromagnetic’ term, a2
could be made strong enough this effect could be weakly reversed.
We have compared the predictions of this model, using the KL theory of interaction, with the observations of
the experimental study that first reported segregation [1]. Using parameter values that fit the osmotic pressure
experiments of [30, 31] and values for the intermolecular densities of these DNA assemblies, as suggested by the
formation of a mild cholesteric state in experiments of [1], we roughly determined the corresponding interaction
parameters of the experiments so that the experimental region could be mapped onto the predicted phase
diagram for mixing/segregation. This mapping revealed a large predicted range for the mixing order parameter
á∣NA - NB∣ñ N, as predicted by the theory, which completely encompasses the range of these values,
á∣NA - NB∣ñ N » 0.1-0.5, found in the experiments.
This indicates that the model is indeed plausible for these weakly cholesteric condensed DNA spherulite
systems, although the large degree in the uncertainty in determining the interaction terms for the experiments
limits comparisons to theory. Unfortunately, again, in those experiments, the exact average value of lattice
spacing R for the spherulites was not measured using x-ray diffraction or some other appropriate method, nor
was the osmotic stress varied with changing PEG concentration. Thus, we cannot better fit our model to
experiments, or vice versa. Therefore, new experiments are clearly needed to determine the dependence of the
degree of phase segregation on R in the spherulites.
We envision that other relatively simple experiments could also be undertaken to study how varying other
molecular parameters of DNA would influence segregation. For example, we would expect that increasing the
length of the molecules would also increase the degree of segregation. Thus, different lengths of DNA fragments
that form a liquid crystal phase should be investigated. Furthermore, we might also expect that changing the
background electrolyte (type or concentration) in these experiments, which would alter the charge environment
of the DNA, could also impact segregation by changing the distribution of adsorbed (or bound) ions about the
DNA . Lastly, one might investigate to what degree changing the base-pair sequence would affect the phase
segregation, and whether this correlates with expected patterns of helical distortions.
These proposed systematic experimental investigations should allow us to better understand the interactions
involved in homology recognition. Indeed, our first goal would be to try to fit the parameters of this current
model to such data, before considering other possible candidate mechanisms of homology recognition
[6, 17, 18]. Notably, such studies would be far-reaching, as the nature of such forces may be important in
initiating critical biological processes such as homologous recombination.
Acknowledgments
AAK is grateful to Japanese Society for promotion of Science (JSPS) for short term award and Prof K Yoshikawa,
his host at Doshisha University, where this work has been started. Discussions with him and Professor Takashi
Ohyama are greatly appreciated. AW acknowledges the Minnesota Supercomputing Institute (MSI) at the
University of Minnesota for providing resources that contributed to the research results reported within this
paper. URL: http://msi.umn.edu.
References
[1] Baldwin G, Brooks N, Robson R, Goldar A, Wynveen A, Leikin S, Seddon J and Kornyshev A A 2008 Duplex DNA recognize sequence
homology in protein free environment J. Phys. Chem. B 112 1060
[2] 2008 Spooky attraction of DNA from a distance New Sci. 15
2008 Paired pairs Nature 451 609
2008 DNA’s self regard Science 319 879
2008 Seeking recognition Biopolymers 89 4
Choi C Q 2008 Double helix double up Sci. Am. 298 30
15
New J. Phys. 19 (2017) 015014
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
D J (O’) Lee et al
Falaschi A 2008 Commentary: Similia similibus: pairing of homologous chromosomes driven by the physicochemical properties of
DNA HFSP J. 2 257
Inoue S, Sugiyama S, Travers A A and Ohyama T 2007 Self-assembly of double stranded DNA molecules at nanomolar concentrations
Biochemistry B 46 164
Danilovicz C, Lee C H, Kim K, Hatch K, Coljee V W, Kleckner N and Prentiss M 2009 Single molecule detection of direct, homologous
DNA/DNA pairing Proc. Natl Acad. Sci. USA 106 19824
Nishikawa J and Ohyama T 2013 Selective association between nucleosome with identical DNA sequences Nucleic Acids Res. 41 1544
Lee D J (O’), Danilowicz C, Rochester C, Kornyshev A A and Prentiss M 2016 Evidence of protein-free homology recognition in
magnetic bead force–extension experiments Proc. R. Soc. A 472 20160186
Cha R S, Weiner B M, Keeney S, Dekker J and Kleckner N 2000 Progression of meiotic DNA replication is regulated by
interchromosomal interaction proteins, negatively by Spo11p and positively by Rec8p Genes Dev. 14 493
Zickler D 2006 From early homologue recognition to synaptonemal complex formation Chromosoma 115 158
Barzel A and Kupiec M 2008 Finding a match: How do homologous sequences get together for recombination? Nat. Rev. Genet. 9 27
Kornyshev A A, Lee D J, Leikin S and Wynveen A 2007 Structure and interactions of biological helices Rev. Mod. Phys. 79 943
Kornyshev A A and Leikin S 1999 Electrostatic zipper motif for DNA aggregation Phys. Rev. Lett. 82 4138
Dickerson R E and Drew H R J 1981 Structure of a B-DNA dodecamer: II influence of base sequence on helix structure J. Mol. Biol.
149 761
Dickerson R E 1992 DNA structure from A to Z Methods Enzymol. 211 67
Kornyshev A and Leikin S 2001 Sequence recognition in the pairing of DNA duplexes Phys. Rev. Lett. 86 366
Kornyshev A A 2010 Physics of DNA: unravelling hidden abilities encoded in the structure of ‘the most important molecule’ Phys.
Chem. Chem. Phys. 39 12352
Lee D J, Wynveen A, Albrecht T and Kornyshev A A 2015 Which way up? Recognition of homologous DNA segments in parallel and
antiparallel alignment J. Chem. Phys. 142 045101
Mazur A K 2016 Homologous pairing between long DNA double helices Phys. Rev. Lett. 116 158101
Lu B, Naji A and Podgornik R 2015 Molecular recognition by van der Waals interaction between polymers with sequence-specific
polarizabilities J. Chem. Phys. 142 214904
Lavery R R, Maddocks J H, Pasi M and Zakrzewska K 2014 Analyzing ion distributions around DNA Nucleic Acids Res. 42 8138
Wynveen A, Lee D J and Kornyshev A A 2005 Statistical mechanics of columnar DNA assemblies Eur. Phys. J. E 16 303
Kornyshev A A and Wynveen A 2009 The homology recognition well as an innate property of DNA structure Proc. Natl Acad. Sci. USA
106 4683
Wynveen A, Lee D J, Kornyshev A A and Leikin S 2008 Helical coherence of DNA in crystals and solution Nucleic Acids Res. 36 5540
Lipfert J, Kerssemakers J W J, Jager T and Dekker N H 2010 Magnetic torque tweezers: measuring torsional stiffness in DNA and RecADNA filaments Nat. Methods 7 977
Smith S B, Cui Y J and Bustamente C 1996 Overstretching B-DNA: the elastic response of individual double-stranded and single-strand
DNA molecules Science 271 5250
Kornyshev A A and Wynveen A 2004 Nonlinear effects in torsional adjustment of interacting DNA Phys. Rev. E 69 041905
Cherstvy A G, Kornyshev A A and Leikin S 2004 Torsional deformation of double helix in interaction andaggregation of DNA J. Phys.
Chem. B 108 6508
Lee D J, Wynveen A and Kornyshev A A 2004 DNA–DNA interaction beyond the ground state Phys. Rev. E 70 051913
Lee D J and Wyneveen A 2006 Torsional fluctuations in columnar DNA assembles J. Phys.: Condens. Matter 18 787
Mezard M, Parisi G and Virosoro M 1987 Spin Glass Theory and Beyond (Singapore: World Scientific)
Rau D C and Paresgian V A 1992 Direct measurement of the intermolecular forces between counterion-condensed DNA double
helices. Evidence for long range attractive hydration forces Biophys. J. 61 246
Rau D C and Paresgian V A 1992 Direct measurement of temperature-dependent solvation forces between DNA double helices
Biophys. J. 61 260
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